I would say no; the example is when $R$ is a hyperbolic plane, the relation you demand says merely that $Q_1$ and $Q_2$ are in the same genus. This is in SPLAG, page 378 in the first (1988) edition, see also Clark Jagy.
Give me a few minutes, I am going to display integer equivalence for $Q_1 = x^2 + 14 y^2,$ $Q_2 = 2 x^2 + 7 y^2,$ which pair take up a good deal of room in Cox's book, and $R = 2 z w.$ By solving $P^T A_1P = A_2$ in 4 by 4 integer matrices...
$$ A_1 = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 14 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right) $$
$$ P = \left( \begin{array}{rrrr} 2 & 7 & -2 & 2 \\ 1 & 5 & -2 & 1 \\ -4 & -14 & 6 & -3 \\ 2 & 14 & -5 & 3 \end{array} \right) $$
$$ A_2 = \left( \begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 7 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right) $$
check with gp-pari
=====================
parisize = 4000000, primelimit = 500509
? a1 = [ 1,0,0,0; 0,14,0,0; 0,0,0,1; 0,0,1,0]
%1 =
[1 0 0 0]
[0 14 0 0]
[0 0 0 1]
[0 0 1 0]
? a2 = [ 2,0,0,0; 0,7,0,0; 0,0,0,1; 0,0,1,0]
%2 =
[2 0 0 0]
[0 7 0 0]
[0 0 0 1]
[0 0 1 0]
? p = [ 2,7,-2,2; 1,5,-2,1; -4,-14,6,-3; 2,14,-5,3]
%3 =
[2 7 -2 2]
[1 5 -2 1]
[-4 -14 6 -3]
[2 14 -5 3]
? matdet(p)
%4 = 1
? pt = mattranspose(p)
%5 =
[2 1 -4 2]
[7 5 -14 14]
[-2 -2 6 -5]
[2 1 -3 3]
? a1
%6 =
[1 0 0 0]
[0 14 0 0]
[0 0 0 1]
[0 0 1 0]
? pt * a1 * p
%7 =
[2 0 0 0]
[0 7 0 0]
[0 0 0 1]
[0 0 1 0]
? a2
%8 =
[2 0 0 0]
[0 7 0 0]
[0 0 0 1]
[0 0 1 0]
? pt * a1 * p - a2
%9 =
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
?
===================